ALBANY—New York Mayor Bill de Blasio’s trip to Albany on Wednesday sparked a fight with Gov. Andrew Cuomo, who threw cold water on many of the city’s top priorities at the Capitol.

After the two leaders met for almost an hour, both said the meeting was cordial. But later Wednesday they aired a sharp disagreement over the mayor’s strategy in Albany, where Mr. de Blasio sought to push for permanent mayoral control over city schools and...